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I've got a couple of non-significant logistic regression models, and am figuring out how to word it appropriately. Before that, I want to confirm the implication. Most questions/websites I have gone to don't go to the verbiage aspect or explain in-depth the implications, hence I'm clarifying here:

If my overall logistic regression model is non-significant: Does this mean: (a) All odds ratios are nonsignificant, or (b) All odds ratios are equal to 1? (i.e., same as the base case)

Thanks for your clarification.

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    $\begingroup$ What would be your interpretation in the linear regression case? $\endgroup$
    – Dave
    Apr 8, 2022 at 16:09
  • $\begingroup$ No opinion - someone (online) once mentioned that it means all coefficients are equal to zero; I took it from there and changed it to '1' in the case of logistic regression, but I'm starting to second guess this response (it could be non-zero, or non-one, and still nonsignificant). Hence my question. $\endgroup$
    – lionclw
    Apr 8, 2022 at 16:10
  • $\begingroup$ Why did you change from $0$ to $1?$ // What is the null hypothesis being tested and coming back as "insignificant"? What is the alternative hypothesis? $\endgroup$
    – Dave
    Apr 8, 2022 at 16:12
  • $\begingroup$ 1. In logistic regression, coefficients for x-values can be converted to odds ratios for ease of interpretation, via the formula exp(coeff). Assuming (b) in my question is the 'correct interpretation, all coefficients are 0, and therefore all odds ratios are exp(0) = 1 Re: your second question: that's the question I'm asking - I'm referring to the p-value generated for the overall regression model (in my stats software), and asking what the null hypothesis is. $\endgroup$
    – lionclw
    Apr 8, 2022 at 16:17

1 Answer 1

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For some link function $g$, let the GLM model be $ g\left(\mathbb E\left[Y\vert X\right]\right) = \beta_0 + \beta_1x_1 +\cdots +\beta_px_p $.

When you get an "overall significance: of the model, the standard hypothesis test is:

$$ H_0: \beta_1 = \cdots = \beta_p = 0\\ H_a: H_0\text{ is false} $$

When you get an "insignificant" result of this hypothesis test, it means that there is insifficient evidence to reject $H_0$ in favor of $H_a$, so insufficient evidence to reject the null hypothesis of all non-intercept parameters being zero.

In your logistic regression, changing a feature by some amount makes no statistically significant change in the log-odds. This is equivalent to saying that the odds ratios are not statistically significantly different from $1$.

$$ \log\left(\dfrac{p_1}{1-p_1}\right) = \log\left(\dfrac{p_2}{1-p_2}\right)\\ \log\left(\dfrac{p_1}{1-p_1}\right) - \log\left(\dfrac{p_2}{1-p_2}\right)=0\\ \log\left(\dfrac{\dfrac{p_1}{1-p_1}}{\dfrac{p_2}{1-p_2}}\right) = 0 \\ \dfrac{\dfrac{p_1}{1-p_1}}{\dfrac{p_2}{1-p_2}} = 1 $$

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  • $\begingroup$ Just keep in mind the usual caveats about an "insignificant" hypothesis test. $\endgroup$
    – Dave
    Apr 8, 2022 at 16:28

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